Metabelian group
From Wikipedia, the free encyclopedia
In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the quotient group G/A is abelian.
Subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomorphisms.
Metabelian groups are solvable. In fact, they are precisely the solvable groups of derived length at most 2.
[edit] Examples
- Any dihedral group is metabelian, as it has a cyclic normal subgroup of index 2. More generally, any generalized dihedral group is metabelian, as it has an abelian normal subgroup of index 2.
- If Fq is a finite field with q elements, the group of affine maps
(where a ≠ 0) acting on Fq is metabelian. Here the abelian normal subgroup is the group of pure translations 
(a group of order q ), its abelian quotient group is isomorphic to the group of homotheties 
(a cyclic group of order q − 1 ).
- The finite Heisenberg group H3,p of order p3 (see the third example Heisenberg group modulo p in the examples section) is metabelian. The same is true for any Heisenberg group defined over a ring (group of upper-triangular 3 × 3 matrices with entries in a commutative ring).
- The symmetric group on four letters S4 is solvable but is not metabelian because its commutator subgroup is the alternating group A4 which is not abelian.
[edit] External links
- Ryan J. Wisnesky, Solvable groups (subsection Metabelian Groups)
